AN ALGORITHM FOR THE FUNDAMENTAL GROUP OF REGULAR …hamilton.nuigalway.ie/preprints/FundGrpAlgo-v10.pdf · 1. Introduction The aim of this paper is twofold. We present the mathematical

AN ALGORITHM FOR THE FUNDAMENTAL GROUP OF

REGULAR CW COMPLEXES.

PIOTR BRENDEL, GRAHAM ELLIS, MATEUSZ JUDA, AND MARIAN MROZEK

Abstract. We describe an algorithm for computing a presentation ofthe fundamental group of a regular CW complex. The algorithm usestechniques from Forman’s combinatorial version of Morse theory and isable to produce small presentations for some large CW complexes arisingin applied topology. To illustrate the performance of the algorithmwe apply it to 3-dimensional CW complexes representing prime knotcomplements and then use group theoretic invariants of the resultingpresentations to classify all prime knots on up to 14 crossings.

1. Introduction

The aim of this paper is twofold. We present the mathematical details ofthe algorithm for computing a presentation of the fundamental group of afinite regular CW complex which was introduced in [2] and which is based onForman’s combinatorial version of Morse theory [9]. We also present detailsof the performance of a C++ implementation of the algorithm. To assessthe performance we systematically apply the algorithm to spaces arising ascomplements of prime knots on fourteen or fewer crossings and use the re-sulting fundamental group presentation, with standard low-index subgroupprocedures, to distinguish between the knots.

1.1. Prior work. The combinatorially defined edge-path group of a con-nected simplicial complex K, due to Reidemeister, is well-known to be iso-morphic to the fundamental group π1(K) (see [27]). It is also well-knownthat this combinatorial definition and isomorphism extends to connectedregular CW-spaces. In this paper we use the terms edge-path group andfundamental group interchangeably as synonyms. Several authors have de-scribed algorithms for implementing Reidemeister’s edge-path group. Reesand Soicher [26] use spanning trees and redundant relator searches in theirdescription of an algorithm for finding a small finite presentation of theedge-path group of a 2-dimensional combinatorial cell complex; they im-plement their algorithm in GAP [12] for 2-dimensional simplicial clique

2010 Mathematics Subject Classification. Primary: 55-04, 55Q05; Secondary: 57M25,52B99.

Key words and phrases. Fundamental group, edge group, algorithm, discrete Morsetheory, knot.

This research is partially supported by the EU under the Toposys project FP7-ICT-318493-STREP, and by the ESF under the ACAT Research Network Programme.

1

2 P. BRENDEL, G. ELLIS, M. JUDA, AND M. MROZEK

complexes of graphs. Letscher [21] uses spanning trees and Tietze elimina-tion/reduction of relators to compute edge-path groups from the 2-skeletaof simplicial complexes arising from knot complements, the knots being pro-duced from experimental data on protein backbones. Palmieri et al. [25]have implemented the edge-path group of simplicial complexes in Sage [28];the implementation uses the 2-skeleton of the complex and calls gap’s Ti-etze reduction/elimination procedures [12]. Kim et al. [18] describe an al-gorithm for the fundamental groups of 3-dimensional simplicial complexes;their algorithm, which makes use of 3-dimensional cells and the language ofgeneral CW-spaces, is applied to 3-dimensional tetrahedral meshes arisingin computer vision.

1.2. Our contribution. Recall that a combinatorial vector field on a reg-ular CW complex X is a partition V of cells of X into singletons and dou-bletons such that each doubleton consists of a cell and one of its facets,that is, one of its faces of codimension 1. The singletons are called criticalcells, the doubletons are called vectors. The combinatorial vector field isacyclic if the facet digraph of X, with direction reversed on edges in V, isacyclic. The algorithm we propose (Algorithm 3.1) inputs a connected finiteregular CW complex X, constructs a homotopy equivalent CW complex X ′

with potentially fewer cells, and outputs the finite presentation for π1(X)corresponding to the 2-skeleton of X ′. The algorithm runs in linear timewith respect to the number of cells in X (see Theorem 3.2). However, itsusefulness depends on the size of X ′. We have no estimate for the numberof cells in X ′ other than it will be no greater than the number of cells inX. However, from several numerical experiments we see that X ′ is usuallysignificantly smaller than X.

At the heart of the algorithm is Forman’s theorem [9, Corollary 3.5] stat-ing that an acyclic combinatorial vector field V on a regular CW complex Xleads to another, homotopic, CW complex whose cells are in one-to-one cor-respondence with the critical cells of V. In order to justify the correctness ofthe algorithm we rework in an algorithmic, recursive spirit the proof of For-man’s theorem. We actually show that the vectors of V may be quotientedout one by one until no vector is left.

We implemented the algorithm in C++ [4]. The implementation aims atlarge CW complexes arising from real-world data such as protein backbonecomplements or point-cloud data sets. However, in order to systematicallystudy the performance of the algorithm, in this paper we apply it to a familyof small cubical complexes arising as the complements of prime knots givenin terms of efficient arc presentations [19]. We have chosen to work withthis case, because it provides a large source of examples whose complexitycan, to some extent, be measured by the number of crossings in the knots.Recall that in [2] we proposed a computable, algebraic invariant of primeknots and used it to classify, up to mirror image, all prime knots up to 11

FUNDAMENTAL GROUP ALGORITHM 3

crossings. The invariant is constructed from low-index subgroups of the fun-damental group of the complement of the knot with the index not exceeding6. The C++ implementation of our algorithm enables fast computation ofthe presentation of the fundamental group. This lets us increase the classifi-cation in [2] from 801 knots up to 11 crossings to 59937 prime knots up to 14crossings in the tabulation provided by Hoste, Thistlethwaite and Weeks in[16] and available from [19]. The C++ implementation shifted the computa-tional barrier from the presentation of the fundamental group computationsto the algebraic invariant computations. The latter are performed in GAP[12] and HAP [7].

The organization of the paper is as follows. In Section 2 we gather nota-tion and preliminaries. In Section 3 we present a recursive approach to thediscrete Morse theory. In Section 4 we study various reduction techniques.In Section 5 we present the algorithms. In Section 6 we apply the results ofthe paper to knot classification.

2. Preliminaries

2.1. Notation. Let N := {1, 2, 3, . . .} stand for the set of natural numbers,R for the set of reals and let I denote the interval [0, 1]. We write f : X−→◦ Yfor a partial map from X to Y , that is a map defined on a subset dom f ⊂ X,called the domain of f , and such that the set of values of f , denoted im f ,is contained in Y . Given a topological space X we write clA for the closureof A ⊂ X.

For n ∈ N we write

Bn := {x ∈ Rn | ||x|| ≤ 1 },Sn−1 := {x ∈ Rn | ||x|| = 1 },Bn−1− := {x = (x1, x2, . . . xn) ∈ Sn−1 | xn ≤ 0 },

Bn−1+ := {x = (x1, x2, . . . xn) ∈ Sn−1 | xn ≥ 0 },◦Bn := {x ∈ Rn | ||x|| < 1 }.

Note that in particular B1 = [−1, 1] and S0 = {−1, 1}. We extend thisnotation to n = 0 by setting

◦B0 := B0 := {∗},S−1 := ∅,

where ∗ is a unique point not belonging to any other set considered in thispaper.

Let A be any set. An oriented element of A is aε := (a, ε) for ε ∈ {−1, 1}and a ∈ A. We denote the set of oriented elements of A by A. We identifya1 with a and we say that a and a−1 have mutually inverse orientation.

By a path in a topological space X we mean a continuous function θ :B1 = [−1, 1] → X. We identify the oriented path θ1 with θ and θ−1 with


the path

B1 3 t 7→ θ(−t) ∈ X.We refer to θ−1 as the inverse path of θ.

2.2. Directed multigraphs. Recall that a directed multigraph is a quadru-ple G = (V,E, e−1, e+1), where e−1, e+1 : E → V are given maps. The setV is the set of vertices of G, the set E is the set of edges of G and the mapse−1, e+1 assign to each edge its initial and terminal vertex respectively.

A path in G is a sequence of oriented edges η := eε11 eε22 · · · eεnn such that

eεk(ek) = e−εk+1(ek+1) for k = 1, . . . n − 1. The initial vertex of the path η

is e−ε1(e1) and the terminal vertex of the path η is eεn(en). The path η is acycle if the terminal and initial vertices coincide. The inverse of the path ηis η−1 := e−εnn e

−εn−1

n−1 · · · e−ε11 .Let ζ be a path in G with the same initial and terminal vertices as an

edge z ∈ E. Moreover, assume that neither z nor z−1 appear in ζ. Let η bea path in G. A (z, ζ)-substitution in η, denoted Sz,ζ(η), is the path obtainedfrom η by replacing every occurrence of z in η by ζ and every occurrence ofz−1 in η by ζ−1.

2.3. Quotient spaces. We recall from [6] some facts concerning quotienttopological spaces. Given an equivalence relation R on a topological spaceX, we denote by [x]R the equivalence class of x ∈ X and by X/R the set ofall equivalence classes of R. Let κR : X 3 x 7→ [x]R ∈ X/R be the canonicalprojection. We drop the subscript R whenever the relation R is clear fromcontext. The quotient topology on X/R consists of subsets A ⊂ X/R suchthat κ−1(A) is open in X.

Proposition 2.1. [6, Proposition 2.4.14] Assume R is an equivalence re-lation on X and S is an equivalence relation on X/R. Then, X/R/S ishomeomorphic to X/RS, where RS is an equivalence relation on X definedby xRSy if and only if κS(κR(x)) = κS(κR(y)).

Given a partial continuous map f : Y−→◦ X such that dom f is closedin Y we define X ∪f Y as the quotient space (X t Y )/Rf of the disjointunion X tY by the equivalence relation Rf which identifies x ∈ dom f withf(x) ∈ X. For z ∈ X t Y we denote by [z]f the equivalence class of therelation Rf . Note that if f is empty, then X ∪f Y coincides with X t Y .In the special case when Y = Bn and dom f = Sn−1 we say that X ∪f Yresults from X by gluing an n dimensional cell via the attaching map f .

The relation R is called closed if the map κR is closed. The followingproposition is a special case of [6, Proposition 2.4.9].

Proposition 2.2. The relation R is closed if and only if for any open U inX the set

UR := {x ∈ U | [x]R ⊂ U }is open in X. �


The following proposition is an easy consequence of Proposition 2.2

Proposition 2.3. Assume Y ⊂ X is closed and R is a closed equivalencerelation in Y . Then R := R∪ idX is a closed equivalence relation on X. �

Proposition 2.4. Assume X is a Hausdorff topological space and R is aclosed equivalence relation in X with compact equivalence classes. ThenX/R is a Hausdorff space.

Proof: Let [x] 6= [y] be two equivalence classes of R. Since they arecompact and X is a Hausdorff space, we can find disjoint open sets U , V inX such that [x] ⊂ U and [y] ⊂ V . By Proposition 2.2 UR and VR are open inX. It follows that κ(UR), κ(VR) are disjoint, open in X/R. Obviously, theyare neighbourhoods respectively of [x] and [y]. Hence, X/R is a Hausdorffspace. �

Given A ⊂ X we denote by X/A the quotient space X/RA, where

RA := { (x, y) ∈ X2 | x = y or x ∈ A and y ∈ A. }

2.4. Finite CW complexes. In this paper by a CW complex we alwaysmean a finite CW complex. Here we recall its definition. Let K be a Haus-dorff topological space. A finite CW structure on K is a pair (K, {ϕσ}σ∈K)such that K is a finite family of subsets of K and:

(CW0) Each element σ ∈ K is a subset of K homeomorphic to◦Bn for some

n ≥ 0. The subset σ is referred to as an n-cell. The number n iscalled the dimension of σ and denoted |σ|. The dimensions providea filtration Kj :=

⋃{σ ∈ K | |σ| ≤ j }.

(CW1) The family K is a decomposition of K, i.e. K =⋃K and any two

different elements of K are disjoint.

(CW2) Each ϕσ is a continuous map ϕσ : B|σ| → K mapping◦

B|σ| homeo-

morphically onto σ and ϕσ(S|σ|−1) ⊂ K |σ|−1. The map ϕσ is calledthe characteristic map of σ.

A finite CW complex (cf. [5, Section I.3]) is a triple (K,K, {ϕσ}σ∈K) suchthat (K, {ϕσ}σ∈K) is a finite CW structure on K. It is convenient to slightlyabuse the language and refer to K as the CW complex, assuming that theCW structure is given implicitly.

The CW complex is called regular if each ϕσ : B|σ| → K is a homeomor-phism onto its image.

By the boundary of a cell σ we mean the set Bdσ := clσ \σ. We say thata cell τ is a face of a cell σ if τ ∩ clσ 6= ∅. The face is proper if τ 6= σ. Theface is a facet if |τ | = |σ| − 1. A 0-dimensional cell is called a vertex.

We refer to the collection of facets of a cell σ as its combinatorial bound-ary and we denote it by bdσ. It is also convenient to consider the collec-tion of cells whose facet is σ. This collection is called the combinatorialcoboundary of σ and is denoted cbdσ. Note that if σ is a 1-cell, then


bdσ = {ϕ(−1), ϕ(1)}. In the case of a 1-cell these two facets (or one ifϕ(−1) = ϕ(1)) are referred to as the endpoints of σ.

A cell is top-dimensional if it is not a proper face of a higher dimen-sional cell. The CW structure and CW complex are called pure if all top-dimensional cells have the same dimension. We will refer to the closure of atop-dimensional cell as a toplex. It is easily seen that a CW complex is theunion of its toplexes.

A union L :=⋃L of a subfamily L ⊂ K such that L is closed in K is called

a subcomplex of K. A subcomplex is a CW complex with its CW structureconsisting of L with the same characteristic maps as in K. Note that, inparticular, every vertex is a subcomplex. Also,

⋃Kj is a subcomplex, called

the j-skeleton of K.Since by [10, Theorem 1.4.10] the closure of a cell in a regular CW complex

is a subcomplex, we have the following proposition.

Proposition 2.5. If cells σ, τ in a regular CW complex satisfy τ ∩ clσ 6= ∅,then τ ⊂ clσ, i.e. τ is a face of σ. �

As a consequence of Proposition 2.5 we obtain the following proposition.

Proposition 2.6. Assume τ is a proper face of σ in a regular CW complexX. Then τ is a facet of a face of σ.

Proof: Set n := |σ|. Since X is regular, Bdσ is homeomorphic to Sn−1.Let k := |σ| − |τ |. Since τ is a proper face of σ we have k > 0. Weproceed by induction on k. If k = 1 the claim is obvious. Thus, assumek > 1. Let x ∈ τ and consider a shrinking sequence of open balls in Bdσ

centered at x. All these balls are homeomorphic to◦

Bn−1 and none of theseballs is contained in τ , because otherwise |τ | = n − 1 and k = 1. Thus,there exists a sequence {xj} ⊂ Bdσ \ τ converging to x. By passing to asubsequence, if necessary, we may assume that the sequence is contained ina cell σ′ ⊂ Bdσ. It follows that τ ∩ clσ′ 6= ∅ and, by the regularity of X, τis a face of σ′. The conclusion now follows from the induction assumption,because |σ′| − |τ | < k. �

Proposition 2.7. ([10, Proposition 1.5.17]) A finite CW complex is a metriz-able topological space. �

Note that if h : Bn → Bn is a homeomorphism, then h maps homeo-

morphically◦Bn onto

◦Bn and Sn−1 onto Sn−1. In consequence, if ϕσ is a

characteristic map, then so is ϕσ ◦ h. Thus, we can freely substitute home-omorphisms into characteristic maps without loosing the CW structure. Inparticular, in the characteristic map ϕσ we can replace Bn by any set home-omorphic to Bn.

Let σ be a top-dimensional cell of K. Then K \ σ is easily seen to be a

subcomplex of K and K results from K \σ by gluing B|σ| via the attachingmap θσ := ϕσ|S|σ|−1 .


Let σ = (σj)kj=1 be an admissible ordering of the cells of K, i.e. an ordering

of all cells in K such that each cell is preceded by its faces. For j = 1, 2, . . . kset

Σj :=

j⋃i=1

σi.

It is not difficult to prove that each Σj is closed in K, hence a subcomplexof K and Σj results from Σj−1 by gluing the cell σj via the attaching mapθσj = ϕ

σj |S|σj |−1 .

One easily verifies that an admissible ordering always exists on a finiteCW complex. Moreover, when L is a subcomplex of K, then one can choosethe ordering in such a way that L = Σl for some l ≤ k. Thus, we have thefollowing standard result.

Proposition 2.8. Each finite CW-complex can be constructed recursivelyby attaching a cell to a subcomplex. �

We will use also the following fact which is a special case of [20, Theorem11.11].

Theorem 2.9. Assume X1, X2 are homotopy equivalent spaces and h :X1 → X2 is a homotopy equivalence. If maps fi : Sn−1 → Xi are suchthat f2 = h ◦ f1, then the spaces X1 ∪f1 Bn and X2 ∪f2 Bn are homotopyequivalent. �

2.5. Oriented CW complexes. Our fundamental group algorithm of aCW complex X is based on a theorem reducing the computation to a groupgenerated by the 1-cells with relators provided by the 2-cells. To presentthe theorem in detail, we recall the definition of the orientation of an n-cellσ in X. For n = 0 it is just an element of {−1, 1}. For n > 0 it is anequivalence class in the homotopy equivalence relation of the characteristicmap ϕσ : (Bn, Sn−1) → (clσ,Bdσ) considered as a map of pairs. By [13,Corollary 2.5.2] there are precisely two orientations for each cell. An orientedCW complex is a CW complex with a fixed orientation for every its cell.

Assume K is an oriented 2-dimensional CW complex. By an edge in K wemean an oriented 1-cell. An edge τ , as each 1-cell, has two endpoints ϕτ (−1)and ϕτ (1). The orientation allows us to distinguish between them. The firstis called the initial vertex of τ and denoted e−(τ), the other is called theterminal vertex of τ and denoted e+(τ). Thus, the CW structure of the 1-skeleton of K may be considered as a directed multigraph. A characteristicmap of an edge is a characteristic map ϕτ of the associated 1-cell whichinduces the chosen orientation of the edge.

Consider now a 2-cell σ. There is a closed path τ1, τ2, . . . , τn in the directedmultigraph of edges such that the homotopy class of the attaching mapθσ contains a map θ : S1 → K1 such that S1 =

⋃nj=1 Ij and θ|Ij is the

characteristic map of the edge τj (see [13, Proposition 3.1.1 and 3.1.2]). Theloop τ1, τ2, . . . , τn is called the homotopical boundary of σ. We have thefollowing theorem.


Theorem 2.10. ([13, Proposition 3.1.7 and Theorem 3.1.8]) Assume K isa connected CW complex with precisely one vertex. Then, the fundamentalgroup of K depends only on the 2-skeleton of K. Moreover, up to an iso-morphism it is the group generated by the edges of K with arbitrarily selectedorientation of K and homotopical boundaries of all 2-cells as relators.

3. Discrete Morse theory

3.1. Reduction pairs. We say that a facet α0 of an n-cell α1 is regular ifϕα1 , the characteristic map of α1, satisfies

(i) ϕα1(Sn−1) is a subcomplex of K,(ii) ϕα1|Bn− is the characteristic map of α0,

(iii) ϕα1(Bn+) ∩ α0 = ∅.

Note that in a regular CW complex each facet is automatically regular.We say that α := (α0, α1) is a reduction pair in a CW complex K if α0 is

a regular facet of α1. The following proposition is straightforward.

Proposition 3.1. If L is a subcomplex of K and α is a reduction pair inL, then α is a reduction pair in K. �

3.2. Discrete vector fields. A discrete vector field V on the CW structureK is a partition ofK into singletons and doubletons such that each doubleton,when ordered by dimension, forms a reduction pair. By a critical cell of Vwe mean a cell α ∈ K such that {α} is a singleton in V. We denote the setof critical cells of V by critV. By pairing the lower dimensional cell with thehigher dimensional cell in every doubleton of V we may and will consider Vas an injective partial self-map V : K−→◦ K such that K is the disjoint unionof domV, imV and critV and for each α ∈ domV the pair (α,V(α)) is areduction pair. We will call such a reduction pair a vector of V.

With V we associate a directed graph, GV . Its vertices are the cells ofK and edges go from each cell σ to each of its facets τ with orientationreversed if σ = V(τ). If GV is acyclic, the discrete vector field V is calledacyclic. In this case the graph GV induces a partial order on the cells in K.In particular, for each cell σ ∈ K there is a minimal cell less than or equalto σ in this partial order. We say about such a cell that it is inferior to σ.

Assume now that K is a regular CW complex with a CW structure K.An algorithm constructing an acyclic discrete vector field on K is presentedin Table 1. It is a simplified version of the algorithm proposed in [14, 15],based on the method of coreductions [23].

For a CW structure K let M(K) denote the maximal cardinality of bdσand cbdσ over all σ ∈ K.

Theorem 3.2. Assume Algorithm 3.1 in Table 1 is called with K containinga CW structure of a non-empty, regular, connected finite CW complex K.The algorithm always stops, returning an acyclic vector field on K. More-over, on return C contains the respective set of critical cells with preciselyone critical cell in dimension zero. If the data structure used to store the


Algorithm 3.1. discreteVectorField(CW structure K)V := L := C := ∅;while K 6= ∅ do

if L = ∅ thenn := min { q | Kq 6= ∅ };move an element r from Kn to C;for each u ∈ (K ∩ cbd r) \ L do enqueue u in L;

elseσ:=dequeue(L);if K ∩ bdσ = {τ} then

remove τ and σ from K and insert (τ, σ) to V ;for each u ∈ (K ∩ cbd τ) \ L do enqueue u in L;

else if K ∩ bdσ = ∅ thenfor each u ∈ (K ∩ cbdσ) \ L do enqueue u in L;

return V;

Table 1. Discrete Vector Field by coreduction method

CW complex provides iteration over the boundary and coboundary of a cell aswell as removing a cell in O(M) time with M := M(K), then the algorithmruns in O(M2 cardK) time.

Proof: Observe that if an element τ ∈ K contributes its coboundaryelements to L as an element of a coreduction pair (τ, σ), then it is removedfrom K, so it may never again contribute in this role. Similarly, if an elementσ with zero boundary contributes its coboundary elements to L, it may neveragain do so, because it is not in the coboundary of any element, so it may notreappear in L. Therefore, each element of S may appear in L at most 2Mtimes. It follows that the while loop is passed at most 2M cardK times. Inparticular, the algorithm always stops and its complexity is O(M2 cardK).

Let Kj , Cj and Vj denote respectively the contents of variable K, C andV on leaving the jth pass of the while loop for j > 0 and the initial valuesof these variables for j = 0. We will show by induction in j that Vj is anacyclic vector field on K with Kj ∪ Cj as the set of its critical cells. This isobvious for j = 0, because V0 = ∅, so every cell is critical. Thus, assume theclaim holds for k < j. It is obvious that Vj is a discrete vector field on K.We only need to prove that Vj is acyclic. This is evident if Vj = Vj−1, thusassume Vj 6= Vj−1. Then Vj = Vj−1∪{(τ, σ)} for some reduction pair (τ, σ).Assume Vj is not acyclic. Then, there exists a cycle s0, t0, s1, t1, . . . sn, tn, s0in GVj such that (ti, si) ∈ Vj and ti is a facet of s(i+1) mod n. In particular,tn ∈ bd s0. Since by induction assumption Vj−1 is acyclic, precisely one ofthe pairs (ti, si) is the pair appearing in Vj on the jth pass of the whileloop. Without loss of generality we may assume that this is the pair (t0, s0).


Thus, we have Kj−1 ∩ bd s0 = {t0}. In particular, t0 ∈ Kj−1. Since the pair(t1, s1) ∈ Vj−1, it must have been included in the vector field on an earlierpass of the while loop, say kth pass with k < j. Thus, Kk−1 ∩bd s1 = {t1}.But t0 ∈ bd s1, therefore t0 6∈ Kk−1. However, the sequence (Ki) is clearlydecreasing, hence t0 6∈ Kj−1, a contradiction. This proves that Vj is acyclicfor every j. In particular, the vector field returned by the algorithm isacyclic. Since after the last pass of the while loop K = ∅, its critical cellsare all stored in C.

We still need to prove that C contains precisely one vertex. First observethat for each j the union of the cells in set Aj := Cj ∪ domVj ∪ imVjis a closed subcomplex of K. Indeed, this is obviously true for j = 0,because A0 = ∅. Now, arguing by induction, we see that Aj+1 \Aj is eitherempty, or a critical cell in Cj+1 whose boundary is in Aj , or a doubleton{τ, σ} such that (τ, σ) is a reduction pair satisfying Kj ∩ bdσ = {τ}, i.e.bdσ \Aj = {τ}. Thus, in each case

⋃Aj is closed, because

⋃Aj−1 is closed

by induction argument.Observe that on the very first pass of the while loop a vertex, say v,

is removed form K and moved to C. Thus, we only need to show that thisnever happens again. Assume the contrary. Let w be another vertex movedto C and assume this happens on the jth pass of the while loop. SinceK is connected, there is a path v = v0, e0, v1, e1, . . . em, vm = w such thateach ei is a 1-cell with endpoints in vertices vi and vi+1. Without loss ofgenerality we may assume that vi ∈ C implies vi = v or vi = w. Since⋃Aj−1 is closed and vm 6∈ Aj−1, we see that also em 6∈ Aj−1. We will

show that also vm−1 6∈ Aj−1. If not, then em would have been placed onL and then removed together with vm as a reduction pair before the jthpass of the while loop. Arguing by induction, we see that v0 6∈ Aj−1, acontradiction. �

3.3. Quotients of reduction pairs. For a reduction pair α we write Kα :=

clα1, ϕα := ϕα1 , |α| := |α1|, K−α := ϕα(Bn−), K+

α := ϕα(Bn+), K

\α :=

K \ α1 \ α0. Set

Kα := { τ ∈ K | τ ⊂ Kα },K+α := Kα \ {α0, α1}.

Lemma 3.3. We have Kα =⋃Kα and K+

α =⋃K+α . In particular, Kα

and K+α are subcomplexes of K.

Proof: Obviously⋃Kα ⊂ Kα. To prove the opposite inclusion take

an x ∈ Kα. If x ∈ α1, then x ∈⋃Kα, because α1 ⊂ Kα. Otherwise,

x ∈ ϕα1(S|α|−1). By assumption (i) of the definition of a regular facet we

have x ∈ τ for some cell τ ⊂ ϕα1(S|α|−1). But ϕα1(S|α|−1) ⊂ Kα, hencex ∈

⋃Kα. This proves the first equality.

We have

Kα = ϕα(B|α|) = ϕα(◦

B|α|) ∪ ϕα(B|α|− ) ∪ ϕα(B

|α|+ ) = α1 ∪ α0 ∪K+

α .


But α1 ∩ K+α = ∅ by the definition of a cell and α0 ∩ K+

α = ∅ by thedefinition of the reduction pair. Therefore,

K+α = Kα \ α1 \ α0 =

⋃K+α ,

which shows that Kα is a subcomplex of K. �Let χ = (χ1, χ2) : B|α| → B|α|−1 × I be a homeomorphism such that

χ(B|α|− ) = B|α|−1 × {1} ∪ S|α|−1 × I,

χ(B|α|+ ) = B|α|−1 × {0}.

Set χα := ϕα ◦ χ−1.Define the map fα : Kα → K+

α by

fα(x) :=

{χα(χ1(ϕ

−1α (x)), 0) if x ∈ α1 ∪ α0,

x otherwise.

It is not difficult to check that fα is well defined and continuous.

Proposition 3.4. The map fα is a retraction of Kα onto K+α , i.e. fα|K+

α=

idK+α

. Moreover, it is a homotopy inverse of the inclusion map K+α ⊂ Kα.

Proof: The fact that fα is a retraction is an elementary calculation. Tosee that fα is a homotopy equivalence, define a homotopy h : Kα× I → K+

α

by

h(x, t) :=

{χα(χ1(ϕ

−1α (x)), tχ2(ϕ

−1α (x))) if x ∈ α1 ∪ α0,

x otherwise.

It is not difficult to check that h is well defined, continuous and h0 = fα,h1 = idK+

α. �

Define an equivalence relation Rα in K by

Rα := { (x, y) ∈ K2 | x = y or x, y ∈ Kα and fα(x) = fα(y) }.Let [x]α denote the equivalence class of x with respect to Rα and let κα :K 3 x 7→ [x]α ∈ K/Rα denote the quotient map.

Proposition 3.5. The space K/Rα is Hausdorff.

Proof: Note that [x]α = f−1α (fα(x)), therefore the equivalence classesof R are compact. Since K is Hausdorff and Kα is closed in K, in view ofProposition 2.4 and Proposition 2.3 it is enough to verify that the restrictionof Rα to Kα is closed. By Proposition 2.2 it suffices to show that for any Uopen inKα the set UR is open inKα. Assume this is not true. Then, for somex ∈ UR no neighbourhood V of x satisfies V ⊂ UR. Thus, Proposition 2.7allows us to select a sequence xn → x such that [xn]R 6⊂ U . Let yn ∈ [xn]\U .Passing to a subsequence, if necessary, we may assume that yn → y ∈ K \U .Since yn ∈ [xn], we have fα(xn) = fα(yn). Thus, fα(x) = fα(y). It followsthat [x]R = [y]R. In consequence y ∈ U , a contradiction. �

Set κ+α := κα|K+α

.


Lemma 3.6. The restriction κ\α := κ

α|K\α: K

\α → K/Rα is a continuous

bijection.

Proof: The map κ\α is continuous as a restriction of a continuous map.

To show that κ\α is surjective, take an x ∈ K. If x 6∈ Kα, then κα(x) =

[x]α. Hence, assume x ∈ Kα. Since fα is a retraction onto K+α , we have

fα(x) = fα(fα(x)). Thus [x]α = [fα(x)]α = κ+α (fα(x)), which shows thatκ+α is surjective. Assume in turn that [x1]α = [x2]α for some x1, x2 ∈ K.Then fα(x1) = fα(x2). If either x1 or x2 is not an element of Kα, thenboth classes are singletons, hence x1 = x2. Thus, assume that x1, x2 ∈ K+

α .Since fα is a retraction onto K+

α , we have x1 = fα(x1) = fα(x2) = x2, whichshows that κ+α is injective. �

Lemma 3.7. The restrictions κα|K+α

and κα|K\Kα are homeomorphisms

onto their images.

Proof: By Lemma 3.6 both maps are continuous bijections. Since Kα iscompact and K/Rα is Hausdorff, the first restriction is a homeomorphismonto K+

α . To see that the other restriction is a homeomorphism observe thatequivalence classes in K \Kα are singletons, therefore κ−1α (κα(U)) = U forany U ⊂ K\Kα. In particular, for any open set U ⊂ K\Kα its image κα(U)is open, which shows that also the other restriction is a homeomorphism. �

3.4. Collapses. We call the quotient space K/Rα an α-collapse of K andwrite briefly κα for the quotient map κRα .

By performing a collapse we do not change the homotopy type and stayin the class of CW complexes, as the following theorem shows.

Theorem 3.8. The quotient space K/Rα is a CW-complex with CW struc-ture ({κα(σ)}σ∈K\{α1,α0}, ({κα ◦ ϕσ}σ∈K\{α1,α0}). Moreover, the complexesK and K/Rα are homotopy equivalent.

Proof: First recall that by Proposition 3.5 K/Rα is a Hausdorff space.Let σ ∈ K \ {α1, α0}. In order to prove property (CW0) of the definitionof CW complex it suffices to show that κα(σ) is homeomorphic to σ. ByLemma 3.3 the set K+

α is a subcomplex, therefore either σ ⊂ K+α or σ ⊂

K \Kα. Thus, by Lemma 3.7, in both cases κα(σ) is homeomorphic to σ,as required. Properties (CW1) and (CW2) are straightforward.

By Proposition 2.8 complex K may be obtained from Kα by consecutivelygluing a sequence (σj) of cells via the attaching maps θσj . It is not diffi-cult to observe that applying to Kα/Rα the same sequence of gluings butwith attaching maps κα ◦ θσj we obtain the complex K/Rα. Therefore, theconclusion follows by induction from Theorem 2.9 and Theorem 3.8. �

Proposition 3.9. If L is a subcomplex of K, then κα(L) is a subcomplexof K/Rα. �


Let V be an acyclic vector field on K. Choose α0, α1 ∈ V such thatα1 = V(α0). Then α := (α0, α1) is a reduction pair. Define a partial mapVα : K/Rα → K/Rα by

domVα := {κα(σ) | σ ∈ domV \ {α0} },Vα(κα(σ)) := κα(V(σ)).

Theorem 3.10. If V is acyclic, then Vα is a well defined discrete vectorfield on K/Rα and it is also acyclic.

Proof: To show that Vα is a well defined discrete vector field on K/Rαconsider β0 ∈ domV \ {α0} and let β1 := V(β0). Then, β := (β1, β0) is

a reduction pair in K. Moreover, β 6= α. Thus, β1, β0 ∈ K\α. Since by

Lemma 3.6 κα is injective on K\α, we see that Vα is injective and domVα ∩

imVα = ∅. Hence, we only need to show that κα(β0) is a regular facetof κα(β1) in Kα. Note that (i) follows from Proposition 3.9 and (ii) from

Lemma 3.6. To prove (iii) assume the contrary. Then, κα(ϕα(B|α|)) ∩κα(β0) 6= ∅. Hence, we can choose an x ∈ K+

β and a y ∈ β0 such that x 6= y

and κα(x) = κα(y). Since κα is bijective on K\α, we have x, y ∈ Kα and

either x or y is not in K+α . But y ∈ α0 ∪ α1 implies β0 = α0 or β0 = α1 and

contradicts α 6= β. Thus, x 6∈ K+α , i.e. either x ∈ α1 or x ∈ α0. Consider

x ∈ α1. Then α1 ∩K+β 6= ∅ which implies Kα ⊂ K+

β and

∅ 6= β0 ∩K+α ⊂ β0 ∩K+

α ⊂ β0 ∩Kα ⊂ β0 ∩K+β ,

a contradiction. Consider x ∈ α0. Then, β0 ∩K+α and α0 ∩K+

β , i.e. α0 is a

facet of β1 and β0 is a facet of α1. Thus, α0, α1, β0, β1, α0 is a cycle in GV ,a contradiction again. This completes the proof that Vα is a well defineddiscrete vector field on K/Rα.

We still need to prove that Vα is acyclic. Assume it is not. Then, we havea cycle

κα(β10), κα(β11), κα(β20), κα(β21), . . . κα(βn0 ), κα(βn1 ), κα(β10)

such that βi := (βi0, βi1) 6= α, κα(βi0)) is a regular facet of κα(βi1) and

κα(βi+10 )) is a facet of κα(βi1). Since βi0, β

i1 ⊂ Kβi and Kβi is a compact

subset of K\α, we see from Lemma 3.6 that κα|Kβi is a homeomorphism,

hence βi0 is a regular facet of βi1. If clβi1 ∩Kα = ∅ for all i, then

(1) β10 , β11 , β

20 , β

21 , . . . β

n0 , β

n1 , β

10

is a cycle in GV , because κα restricted to K \Kα is a homeomorphism ontoits image. Hence, assume that βi+1

0 is not a facet of βi1 for some i but

κα(βi+10 ) ⊂ clκα(βi1). This is only possible if α0 is a facet of βi1 and βi+1

0 is a

facet of α1. Thus, we can insert α0, α1 between βi+10 and βi1 in (1) obtaining

a cycle in GV , a contradiction. �A sequence α = (α1, α2, . . . , αm) of reduction pairs in K is called ad-

missible if the associated collection {α1, α2, . . . , αm} of reduction pairs is a


discrete vector field. Obviously, given a discrete vector field V, any orderingof its elements is admissible. Assuming V is acyclic and fixing an order-ing α, we can use Theorem 3.10 to recursively apply collapses of reductionpairs in α. By Theorem 3.8, the resulting space is a CW complex homotopyequivalent to the original complex K. We call it the Morse complex of thevector field V and denote K/V. This is some abuse of language, becausethe construction depends on the ordering α but we assume the ordering isimplicit. Summarizing, we get the following fundamental result of discreteMorse theory.

Corollary 3.11. Let K be a finite CW complex and V an acyclic discretevector field on the CW structure K of K. Then, the complexes K and K/Vare homotopy equivalent.

�

4. Reductions.

4.1. External collapses. We say that the reduction pair α is free if K\α is

a subcomplex of K.

Theorem 4.1. If α is free then the complexes K and K\α are homotopy

equivalent.

Proof: We know from Lemma 3.6 that the restriction κα|K\α

is a con-

tinuous bijection onto K/Rα. Thus, it is a homeomorphism, because K\α,

as a subcomplex, is compact. Hence, the conclusion follows from Theo-rem 3.8. �

When α is free, we say that K\α is an external collapse of K. Otherwise

we say that K/Rα is an internal collapse of K. On the algorithmic sideexternal collapses are advantageous, because to compute them it is enoughto remove the reduction pair from the list K of cells of K. Thus, it isconvenient to make as many external collapses as possible. We say that Kcollapses onto the subcomplex L if there exists a sequence of subcomplexesL = L0 ⊂ L1 ⊂ . . . ⊂ Lm = K such that Lj−1 is an external collapse of Lj .Note that for convenience we admit the case when m = 0, i.e. L = K.

An immediate consequence of Proposition 3.4 is the following proposition.

Proposition 4.2. If K collapses onto a subcomplex L then the inclusionL ⊂ K is a homotopy equivalence. �

4.2. Collapsible subcomplexes. Complex K is called collapsible if it col-lapses onto a vertex.

Theorem 4.3. Assume L is a collapsible subcomplex of K. Then, K andK/L are homotopy equivalent.

Proof: Let Lm ⊂ Lm−1 ⊂ . . . ⊂ L0 = L be a sequence of subcomplexesof L such that Lj is an external collapse of Lj−1 by a reduction pair αj


and Lm = {∗} consists only of a vertex. Let κj : Lj → Lj/Rαj be thequotient map and let κ∗ := κm ◦ κm−1 ◦ · · ·κ1. By Proposition 3.1 each αj

is a reduction pair in K. Thus, by Theorem 4.1 K is homotopy equivalentto

K/Rα1/Rα2 · · · /Rαmand by Proposition 2.1 this space is homeomorphic to K/R∗, where R∗ isa relation on X defined by xR∗y if and only if κ∗(x) = κ∗(y). But κ∗ isconstant on L and a homeomorphism on K \L. Hence, K/R∗ and K/L arehomeomorphic, and consequently K and K/L are homotopy equivalent. �

Theorem 4.4. Assume K is a regular CW complex and L is a subcomplexwith CW structure L ⊂ K. Then, K collapses onto L if and only if thereexists an acyclic discrete vector field on K such that critV = L.

Proof: Assume first K collapses onto L. Then, we have subcomplexesL = L0 ⊂ L1 ⊂ . . . ⊂ Lm = K such that Lj−1 is an external collapse ofLj resulting from removing a reduction pair αj . It is straightforward toverify that the collection{α1, α2, . . . αn} is an acyclic discrete vector field onK whose set of critical cells is precisely L.

To prove the opposite implication assume V is an acyclic discrete vectorfield on K such that critV = L. Let n be the cardinality of K\L. We proceedby induction on n. For n = 0 the conclusion is trivial. Thus, consider thecase n > 0. Then, we can choose a non-critical cell in σ ∈ K. Since V isacyclic, we can choose a cell τ which is inferior to σ, that is τ is a minimalcell less than or equal to σ in the partial order induced by V. We claim thatτ is not a critical cell of V. To see this consider a path τ = τ0, τ1, . . . , τm = σin GV . Since τ is inferior to σ, such a path obviously exists. If τ is critical,then we consider the smallest index k such that τk−1 is critical and τk is notcritical. Since the partial order induced by GV goes against the facet partialorder only for the vectors of V, we have τk ∈ domV and τk ⊂ cl τ ⊂ L,because τ , as a critical cell is a subset of L and L as a subcomplex is closed.It follows that τk is critical, a contradiction. Therefore τ ∈ domV. LetK ′ := K \ τ \ V(τ). To prove that K ′ is closed, assume the contrary. Then,there exists an x ∈ clK ′ \K ′. Hence, x ∈ τ ∩ V(τ) and x ∈ clσ′ for somecell σ′ ⊂ K ′. By Proposition 2.5 τ is a face of σ′ and by Proposition 2.6τ is a facet of a face σ′′ of σ′. Thus σ′′ precedes τ in the partial orderinduced by GV , and consequently τ is not inferior to σ. This contradictionshows that K ′ is a subcomplex of K and consequently K collapses onto K ′.By induction assumption K ′ collapses onto L. Therefore, K collapses ontoL. �

As an immediate consequence of Theorem 4.4 we obtain the followingcorollary.

Corollary 4.5. A regular CW complex is collapsible if and only if it admitsa discrete vector field whose set of critical cells consists of precisely onevertex. �


4.3. Redundant configurations. Let K be a regular CW complex, L itssubcomplex and K and L the corresponding CW structures. We refer tothe pair (K,L) as a configuration. We say that the configuration (K,L) isredundant if there exists a discrete vector field V on K such that critV = L.We say that subcomplexes L, M of K form a redundant decomposition of Kif K = L ∪M and the configuration (M,L ∩M) is redundant.

Theorem 4.6. Assume subcomplexes L and M of a regular CW complex Kform a redundant decomposition. Then, the inclusion L ⊂ K is a homotopyequivalence.

Proof: Let V be a discrete vector field on M such that critV consists ofthe cells in L∩M . Extend it to all cells in K by making all cells not in thedomain of V critical. It follows from Theorem 4.4 that K collapses onto L,thus we get the assertion from Proposition 4.2. �

The advantage of Theorem 4.6 is twofold. We may use it to reduce thecomplex without changing its homotopy type by removing a subcomplexfrom a redundant decomposition. We may also simplify the complex byquotienting out a maximal collapsible subcomplex. We can construct sucha subcomplex L ⊂ K by starting from a vertex and recursively adding toL a subcomplex M such that L and M form a redundant decomposition ofL ∪M . However, to make it work in practice we need a quick method totest for redundant configurations.

This is possible in the setting of tessellations discussed next.

4.4. Tesselated spaces and lattice complexes. We say that a CW struc-ture K on a pure, regular CW complex X, finite or not, is a tessellation ifthe closures of any two top-dimensional cells σ1, σ2 in K are isomorphic, i.e.there exists a homeomorphism h : clσ1 → clσ2 mapping cells onto cells. ACW complex is a tesselated CW complex if its CW structure is a tessellation.Obviously, every pure subcomplex of a tesselated space is a tesselated space.

A special way to define a tessellation is to consider a lattice L ⊆ Rn, i.e. anadditive subgroup of Rn generated by some choice of n linearly independentvectors VL = {v1, · · · , vn}. Any v ∈ L determines a Dirichlet–Voronoi cell

DL(v) := {x ∈ Rn | ||v − x|| ≤ ||w − x|| ∀w ∈ L}.

For x ∈ Rn set

χL(x) := { v ∈ L | x ∈ DL(v) }and define an equivalence relation R ⊂ Rn × Rn by xRy if and only ifχL(x) = χL(y). One can show that the equivalence classes of this relationprovide a CW structure of Rn which is a tessellation. We call it a latticetessellation and denote it KL. When VL consists of n orthogonal vectors, thesets DL(v) are n-dimensional hypercubes and the tessellation is the classicalcubical decomposition of Rn. By identifying Rn with the hyperplane

{(x1, . . . , xn+1) ∈ Rn+1 : x1 + · · ·+ xn+1 = 0}


and taking as VL the set v1 = (1−n, 1, 1, . . . , 1, 1), v2 = (1, 1−n, 1, . . . , 1, 1),. . . , vn = (1, 1, 1, . . . , 1 − n, 1) we obtain a tessellation of Rn consisting ofn-dimensional permutahedra.

A pure, finite CW complex whose CW structure is a subfamily of a latticetessellation will be called a lattice CW complex. Lattice CW complexes maybe easily and efficiently stored in bitmaps.

Fix a lattice L. Two top-dimensional cells in KL are called neighbours ifthe intersection of their closures is non-empty. Consider a top-dimensionalcell σ ∈ KL and a lattice CW complex K with its CW structure K ⊂ KL.

Observe that K \ σ and clσ are also lattice CW complexes. Their inter-section is a subcomplex of X. We will call it the contact complex of σ andK. Note that this makes sense both if σ ⊂ K and σ ∩K = ∅.

Let NK(σ) denote the set of neighbours of σ contained in K. Then

(K \ σ) ∩ clσ =⋃{ clσ ∩ cl τ | τ ∈ NK(σ), τ 6= σ },

i.e. the contact space is uniquely determined by the neighbours of σ in K.The toplex clσ is said to be redundant in K if the configuration (clσ,K \

σ ∩ clσ) is redundant. If σ has k neighbours, then there are 2k configura-tions to test for redundancy. Since all toplexes in a lattice tessellation areisomorphic, tests done for one toplex may be reused for other toplexes. Evenbetter, for reasonably small number of neighbours we can test all configura-tions of a toplex for redundancy and store them in a lookup table for laterquick redundancy tests. The number of neighbours for a cubical tessellationin Rn is 3n − 1. The number of neighbours for a permutahedral tessellationin Rn is 2n+1 − 2. Thus, the lookup tables may be computed and stored inless than 1GB of memory for cubical tessellations up to dimension 3, requir-ing 226 bits or 223 bytes in dimension 3 and for permutahedral tessellationsup to dimension 4, requiring 230 bits or 227 bytes in dimension 4.

5. Reduction algorithms.

5.1. Shaving and collapsing. Two reduction algorithms based on redun-dancy tests are presented in Tables 2 and 3. The first, often called shaving,removes redundant toplexes as long as they exist. The second constructs apossibly large collapsible subcomplex.

Theorem 5.1. Algorithm 5.1 applied to a collection of toplexes representinga lattice space always stops and returns a homotopy equivalent subcomplexwith no redundant toplexes.

Proof: The conclusion follows immediately from Theorem 4.6. �

Theorem 5.2. Algorithm 5.2 applied to a collection of toplexes represent-ing a lattice space always stops and returns a collapsible subcomplex. Theresulting quotient complex is homotopy equivalent to the complex on input.

Proof: The conclusion follows immediately from Theorem 4.6 and The-orem 4.3. �


Algorithm 5.1. shaving(collection of toplexes K)Q := ∅;while there exists a redundant toplex σ in K do

remove σ from K;enqueue NK(σ) in Q;while Q 6= ∅ do

σ=dequeue(Q);if σ is a redundant in K do

remove σ from K;enqueue NK(σ) in Q;

return K;

Table 2. Shaving

Algorithm 5.2. collapsibleSubset(collection of toplexes K)C := Q := ∅;choose any toplex σ in K and insert it into C;enqueue NK\C(σ) in Q;while Q 6= ∅ do

σ:=dequeue(Q);if σ is redundant in C do

insert σ into C;enqueue NK\C(σ) in Q;

return C;

Table 3. Collapsible subset construction

5.2. C-structures. In order to obtain an algorithm based on acyclic vectorfields and Theorem 2.10 it is convenient to consider the following concept.

Let C be a finite set decomposing into the disjoint union C = C0 ∪C1 ∪ C2. By a C-structure we mean a quadruple (C, e−1, e+1, d) such that(C0, C1, e−1, e+1) is a directed multigraph and d : C2 → Cycles(C1), whereCycles(C1) stands for the set of cycles in C1.

We write |c| = j if and only if c ∈ Cj .For c ∈ C we define its boundary by

bd c :=

∅ if |c| = 0,

{e−1(c), e+1(c)} if |c| = 1,

{ e1, e2, . . . en } if |c| = 2 and d(c) = eε11 eε22 · · · eεnn .


With every C-structure C we associate a 2-dimensional CW complex K

defined as follows. For each c ∈ C we have a formal cell 〈c〉 :=◦B|c| ×{c}.

However, to simplify notation, in the sequel we identify 〈c〉 with c. Let

Kj :=⋃{ c ∈ C | |c| ≤ j }.

For c ∈ C1 and t ∈ B1 we define the characteristic map ϕc : B1 → K1 by

ϕc(t) :=

{(t, c) if t ∈

◦B1,

et(c) otherwise.

To define the characteristic map ϕc : B2 → K2 for a c ∈ C2 assumed(c) = eε11 e

ε22 · · · eεnn and take an x ∈ B2. Set

ϕc(x) :=

{(x, c) if x ∈

◦B2,

ϕεkek( nt2π − k) if x = e2πti and 2πn k ≤ t ≤

2πn (k + 1).

It is not difficult to verify that the cells c together with the characteristicmaps ϕc define a CW complex. We denote it by CW(C). Conversely, given afinite, regular, 2-dimensional CW complex K it is straightforward to obtaina C-structure C such that CW(C) is isomorphic to K.

Given a reduction pair (α0, α1) in CW(C) we define a C-structure C ′ asfollows. Let C ′ := C \ {α0, α1}. To define e′ε : C ′1 → C ′0 and d′ : C ′2 →Cycles(C ′1), consider first the case |α1| = 1. By the definition of a reductionpair α0 is a regular facet of α1. This implies that bdα1 consists of preciselytwo elements one of which is α0. Let α∗0 be the other. Set

e′ε(c) :=

{α∗0 if eε(c) = α0,

eε(c) otherwise,

d′(c) := d(c).

Consider the case |α1| = 2. Since α0 is a regular facet of α1, the facet α0

appears precisely once in the cycle d(α1) = eε11 eε22 · · · eεnn . Without loss of

generality we may assume that α0 = e1. Let α∗0 := (eε22 eε33 · · · eεnn )−ε1 . Then

α∗0 and α0 have the same initial and terminal vertex. Moreover, α0 does notappear in α∗0. Set

d′(c) := Sα0,α∗0(d(c)),

where Sα0,α∗0denotes the α0, α

∗0 substitution in d(c).

It is not difficult to verify that (C ′, e′−1, e′+1, d

′) is a C-structure. We callit the α-collapse of C. One easily verifies the following proposition.

Proposition 5.3. The CW complex CW(C ′) associated with the α-collapseC ′ is homeomorphic to the quotient complex CW(C)/Rα. �

Algorithm 5.3 computing a presentation of the homotopy group of a reg-ular CW complex K represented as a collection of top-dimensional cells ispresented in Table 4.


Algorithm 5.3. fundGroup(collection of toplexes K)K := shaving(K);A := collapsibleSubset(K);C := C-structure of K/A;V := discreteVectorField(C);for each α ∈ V do

assign to C the α-collapse of C;return (C1, d(C2));

Table 4. Fundamental group presentation of a regular CW complex.

Theorem 5.4. Assume Algorithm 5.3 is called with a collection of top cellsof a CW structure of a lattice space K. It always stops and returns a pre-sentation, up to an isomorphism, of the fundamental group of K.

Proof: The theorem follows immediately from Theorem 5.1, Theorem 5.2,Theorem 3.2, Theorem 3.8, Proposition 5.3, Theorem 3.10 and Theorem 2.10.

�

6. Application to knot classification.

6.1. Knot classification. Recall that two knots K,K ′ ⊂ R3 are equivalent[22, 11] if there exists a homeomorphism f : R3 → R3 such that K ′ = f(K).They are orientation equivalent [22] or isotopic [11] if the homeomorphismis an ambient isotopy of knots, that is it is isotopic to the identity. Gordonand Luecke [11, Theorem 1] prove that K,K ′ are equivalent if and only ifthe complements R3 \K and R3 \K ′ are homeomorphic. Thus, topologicalinvariants of the complements of knots may be used to distinguish betweennon-equivalent knots. They cannot be used to distinguish between non-isotopic knots. But, by a result of Fisher [8] two equivalent but non-isotopicknots differ only by the mirror image.

Let T be an invariant of topological spaces, that is T (X) = T (Y ) if X andY are homeomorphic. Then, obviously T (R3 \K) = T (R3 \K ′) if K and K ′

are equivalent. We say that T classifies a family of knots K if for any twonon-equivalent knotsK,K ′ ∈ K we have T (R3\K) 6= T (R3\K ′). Note that ifthe familyK is finite and the invariant T is algorithmically computable, then,at least theoretically, one can verify if T classifies K in finite computation.A natural candidate for a good T is the fundamental group. Unfortunately,by the results of Novikov [24] and Boone [1], in general it is not possibleto decide algorithmically from finite presentations of two groups if they arenon-isomorphic. However, this finite presentation may be used to computesome further, algorithmically decidable, invariants. This is the path we usedin [2]. Recall that given a knot K ⊂ R3, using Algorithm 5.3 we can compute


c 3 4 5 6 7 8 9N(c) 1 1 2 3 7 21 49

c 10 11 12 13 14 15 16N(c) 165 552 2176 9988 46972 253293 1388705

Table 5. The growth of the number of prime knots N(c)with the number of crossings c.

a presentation of the fundamental group GK := π(D3 \MK), where D3 is acubical ball in R3 containing the knot K and MK is a cubical neighbourhoodof K containing K as a deformation retract. Using this presentation andthe GAP software library, we can compute some algebraic invariants of GK .In particular, given integers n, c ≥ 1, m ≥ 0 we study in [2] the invariant

I [n,c,m](K). It is defined by

I [n,c,m](K) := {Hm(S/γc+1S,Z) : S ≤ GK , |GK : S| ≤ n},where Hm(G,Z) denotes the integral homology of group G and γkG is de-fined recursively by γ1G := G, γk+1G := [γkG,G]. Trying to balancethe computational expenses between the cost of computing the invariantI [n,c,m](K) and its ability to distinguish between knots we chose the in-

variant In(K) := I [n,1,1](K) with c = m = 1 and varying n. Using a GAPimplementation of Algorithm 5.3 we provide in [2] a computer assisted proofof the following result.

Theorem 6.1. The invariant I6(K) distinguishes, up to mirror image, be-tween ambient isotopy classes of all prime knots that admit planar diagramswith eleven or fewer crossings. �

6.2. Classifying index. Using a C++ implementation of Algorithm 5.3and an improved strategy of computing the classifying invariant we are ableto strengthen Theorem 6.1 (see Theorem 6.3 below). The computationalchallenge of this extension lies, in particular, in the exponential growth of thenumber N(c) of prime knots with c crossings. Prime knots up to 16 crossingwere tabulated by Hoste, Thistlethwaite and Weeks in [16]. The number ofsuch knots for a given n is presented in Table 5. In our computations weused the Hoste-Thistlethwaite tabulation available from [19].

There are only 801 prime knots up to 11 crossings but 59937 prime knotsup to 14 crossings. The additional challenge is in the cost of computing theinvariant In, particularly when n grows. An observation we made is that formost knots relatively low values of n suffice to distinguish them from otherknots. But, some knots require relatively high values of n. Thus, in orderto decrease the computational cost, it makes sense to take as the classifyinginvariant the pair (n, In(K)) for a possibly low n selected individually foreach knot. Assuming the invariant classifies a family of knots K, given aknot K ∈ K we refer to n in the invariant (n, In(K)) as the classifying index


Algorithm 6.1. classify(collection of knots K)Q := empty queue; // to process (K,n)A := empty set; // (K,n) added to QI := empty dictionary; // I[i] is a list of (K,n), s.t. i = In(K)C := empty dictionary; // C[K] final invariant for KD := empty dictionary; // D[i] = K iff C[K] = iN := 0; // classifying indexforeach K ∈ K do

enqueue (K, 2) in Q;insert (K, 2) into A;

while Q is non-empty do(K,n):=dequeue(Q);N := max(N,n);i := In(K);if i is not a key in I then

I[i] := empt list;append (K,n) to list I[i];D[i] := K;C[K] := (n, i);

elseappend (K,n) to list I[i];remove key i from D;foreach (K, n) ∈ I[i] do

(*) remove K from C;if (K, n+ 1) /∈ A then

(**) enqueue (K, n+ 1) in Q;insert (K, n+ 1) into A;

return (C,D,N);

Table 6. Classification algorithm.

of K. By the classifying index of the family K we mean the maximum ofall classifying indexes of individual knots K ∈ K. The idea of classifyingindexes leads to Algorithm 6.1 presented in Table 6. Note that the whileloop in Algorithm 6.1 is potentially infinite. But, we have the followingtheorem.

Theorem 6.2. Assume Algorithm 6.1 starts with a collection of knots Kon input and stops. Then, the triple (C,D,N) on output, consisting ofdictionaries C, D and an integer N , has the following features.

(a) The integer N is the classifying index of K.(b) The keys of C are all knots in K.


(c) The keys of D constitute a collection of invariants In(K), one pereach knot K ∈ K. Given an invariant i in the keys of D, the as-sociated value D[i] is the unique knot K in K such that C[K] =(n, In(K)) = (n, i). Then, the number n is the classifying index ofK.

(d) Both dictionaries C and D are injective. In particular, for any twoknots K,L ∈ K the invariants C[K] and C[L] are different.

Proof: Given a variable x in Algorithm 6.1 we let x(l) denote the stateof x after l iterations of the while loop. In particular, for l = 0 this is thestate just before entering the while loop. We will show by induction thatfor every l ≥ 0 we have the following properties:

(i) A(l) =⋃lk=0Q

(k),

(ii) the sequence {n(i) }li=1 is not descending,

(iii) (K(l), n(l)) 6= (K(l′), n(l′)) for each l′ < l,

(iv) for each K ∈ K the value of C(l)[K] is set if and only if K ∈ K\L(l),where

L(l) := {K ∈ K | ∃n∈N(K,n) ∈ Q(l) },(v) C(l) and D(l) are injective,

(vi) the value D(l)[i] is the unique K ∈ K such that

(2) C(l)[K] = (n, In(K)) = i.

First observe that dictionaries C(0), D(0), I(0) are empty, A(0) is equal toQ(0) because of the first for loop, Q(0) contains only pairs (K, 2) for every

K ∈ K, L(0) = K, n(0) is not defined, n(1) = 2 and n(j) ≥ 2 for j ≥ 1. Inparticular, properties (i)-(vi) are trivially satisfied for l = 0. Thus, fix l > 0and assume the conditions are satisfied for k < l.

To see (i) observe that in course of the lth iteration of the while loopwe add an element to A if and only if we add it to Q and we never removeelements from A. This proves (i).

By the induction assumption n(k) ≥ n(k−1) for k < l. Hence, in orderto prove (ii) we only need to show that n(l) ≥ n(l−1). The inequality is

obvious when n(l−1) = 2. Thus, assume that n(l−1) > 2. Then, n(l−1)

and consequently also n(l) enter the queue Q in pair with a respective knotinside the while loop. Hence, there exist k and k′ such that l > k > k′,n(l) = n(k) + 1 and n(l−1) = n(k

′) + 1. From the induction assumptionn(k) ≥ n(k′), therefore n(l) = n(k) + 1 ≥ n(k′) + 1 = n(l−1).

To prove (iii) let l′ < l be such that (K(l), n(l)) = (K(l′), n(l′)). This

obviously cannot happen if n(l′) = 2. Thus, for l there exists a k < l and for

l′ there exists a k′ < l′ such that

(K(l), n(l)) ∈ Q(k) \Q(l)

and

(K(l′), n(l′)) ∈ Q(k′) \Q(l′).


Without loss of generality we may assume that k′ < k. In the kth iterationline (∗∗) is executed with (K(l), n(l)) /∈ A(k). However,

(K(l), n(l)) = (K(l′), n(l′)) ∈ Q(k′) ⊂ A(k′) ⊂ A(k),

a contradiction.Now we prove (iv). Assume to the contrary that there exists a K ∈ K\K(l)

such that C(l)[K] is not set. Then, on the lth pass of the while loop line (∗)is executed with K(l) = K. Indeed, if K 6∈ L(l−1), then C(l−1)[K] is set and

the algorithm executes line (∗) by the induction assumption. If K ∈ L(l−1),then K = K(l) and the algorithm executes the else branch, because C(l)[K]

is not set. In particular, (K,n) with n = n(l) is processed as one of the

elements of I(l)[i] in the second foreach loop. If the algorithm executes

line (∗∗), then K ∈ K(l), a contradiction. Otherwise (K,n+ 1) ∈ A(l−1) and

consequently (K,n + 1) ∈ Q(l′) for some l′ < l. Because of the assumption

(K,n+ 1) also is not in Q(l). Hence, K = K(l′′) and n+ 1 = n(l′′) > n(l) = n

for some l′′ such that l′ < l′′ < l. This contradicts (ii).

Assume in turn that C(l)[K] is set and K ∈ L(l). If C(l−1)[K] is set, then

by the induction assumption K 6∈ L(l−1). Thus, the algorithm executes line(∗∗) in the lth iteration with K1 = K. In particular, it also executes line

(∗), a contradiction. If C(l−1)[K] is not set, then K(l) = K. Let n := n(l).

Since K ∈ L(l) and the algorithm stops, there is a k > l such that K(k) =K. Without loss of generality we may assume that k is minimal with thisproperty. By (ii) we have n(k) ≥ n(l) and by (iii) n(k) > n(l). It cannot be

n(k) = n(l) + 1, because on the lth pass of the while loop the else branchis not executed. Thus, n(k) > n(l) + 1. It follows that there exists a p < ksuch that K(p) = K and n(k) > n(p) + 1. Therefore, n(p) > n(l) and by (ii)p > l. Consequently, the number k, contrary to its choice, is not minimal.

Now we prove (v). It follows easily from the induction assumption that

if C(l) is not injective, then there exists a k < l such that

C(l)[K(l)] = C(l)[K(k)] = C(k)[K(k)].

It means that

(n(l), i(l)) = (n(k), i(k)).

We set C[K] only in the then branch of the first if statement. Hence,

i(l) /∈ I(l−1). But in the kth iteration of the while loop we set C[K(k)].

Therefore, i(k) ∈ I(k). We do not delete elements from I. Thus, sincek ≤ l − 1, we have i(l) = i(k) ∈ I(l−1), a contradiction.

Again from the induction assumptions, if D(l) is not injective, then thereexists a k < l such that

K(l) = D(l)[i(l)] = D(l)[i(k)] = K(k),

and i(l) 6= i(k). Since K(l) = K(k), we have i(l) = In(K(l)) = In(K(k)) = i(k),a contradiction.


c 3 4 5 6 7 8 9 10 11 12 13 14N(c) 2 2 3 3 3 3 5 5 6 6 7 7

Table 7. Classifying indexes N(c) for the family HTW≤cwith c ≤ 14.

To prove (vi) let i,K be such that D(l)[i] = K. Since D is injective, we see

that K is unique. If D(l)[i] = D(l−1)[i] then (2) follows from the inductionassumption. Otherwise, the algorithm sets D in the lth iteration. Thus, italso sets C[K] to (n, i).

To finish the proof first observe that assertion (a) is obvious. Let l? denote

the final pass of the while loop. Then, Q(l?) = ∅. Thus, we get from (iv)that the keys of C are all knots in K. This proves assertion (b). Assertion(c) follows from (vi) and assertion (d) from (v). �

Given a family K of prime knots and an integer c we denote by Kc the sub-family of these knots in K whose minimal planar diagram requires c crossingsand we set K≤c :=

⋃i≤cKi. Let HTW denote the family of 1 701 936 prime

knots tabulated by Hoste, Thistlethwaite and Weeks in [16] and availablefrom [19]. We have the following extension of Theorem 6.1.

Theorem 6.3. The invariant In(K) distinguishes, up to mirror image, be-tween ambient isotopy classes of all prime knots in HTW≤14 and the clas-sifying index of this family is 7.

Proof: Apply Algorithm 6.1 to the family HTW≤14. The algorithm stops,returning 7 as the classifying index of the family. Hence the result followsfrom Theorem 6.1. �

The same way we can determine the classifying index for HTW≤c withc ≤ 14. The results are presented in Table 7. Figure 1 presents the groupingof classifying indexes of individual knots. Tables 8, 9 and 10 indicate thatfor most knots their classifying index is relatively low.

6.3. Limitations of the method. Recall that during the computationsof the In invariant of a knot K we first build a finite presentation of thefundamental group G of R3 \ K and construct the family Sn(G) of thesubgroups of index n of G.

Since each subgroup of index n of a finitely generated group G correspondsto a homomorphism G→ Sn, the number of such subgroups is bounded fromabove by (n!)g where g is the number of generators in G. We cannot boundthe number from below. Thus, we can influence the computational timeof constructing Sn(G) only via g. Therefore, since g is not greater thanthe number of critical cells of the constructed discrete vector fields, it isimportant to minimize this number. The problem of computing Sn(G) iswell studied in the literature. A good source of references is [3].


Figure 1. The image shows knots separation in HTW≤14 bythe invariant In. The middle white circle represents all knotsin HTW≤10. The first ring counting from inside presents arcsgrouping knots with the same invariant with classifying index2. The length of the coloured arc represents the group size.The following rings show groups for the classifying indexes 3,4 and 5. We can see how the groups split. Also, it is visiblethat the classifying index exceeds 3 only for a small numberof knots.

The problem of computing the optimal number of critical cells is NP-complete [17]. In the construction of the discrete vector field we use greedyalgorithms. They usually behave very well for homology computation, wherethe difference of one more or less critical cell has no significant consequencefor the total computational complexity. However, in the case of the I7 in-variant, one more critical cell may increase the total computation time fromminutes to days. The most time-consuming knot we encountered is K14,38437

presented in Figure 2. For this knot the computations of I7(K) vary betweena few minutes and 52 days, depending on the method used to construct thediscrete vector field (coreductions, reductions) and the order of cells in thedata structure. The best method and order varies from knot to knot. Since


c\n 2 3 4 5 6

3 1 (100%) 1 (100%) 0 (0%) 0 (0%) 0 (0%)4 1 (100%) 1 (100%) 1 (100%) 0 (0%) 0 (0%)5 2 (100%) 2 (100%) 2 (100%) 2 (100%) 0 (0%)

6 3 (100%) 3 (100%) 3 (100%) 2 (67%) 0 (0%)7 7 (100%) 7 (100%) 6 (86%) 3 (43%) 0 (0%)8 21 (100%) 21 (100%) 14 (67%) 10 (48%) 1 (5%)

9 49 (100%) 49 (100%) 30 (61%) 19 (39%) 1 (2%)10 165 (100%) 165 (100%) 110 (67%) 75 (45%) 1 (1%)11 552 (100%) 552 (100%) 355 (64%) 225 (41%) 10 (2%)

12 2176 (100%) 2156 (99%) 1151 (53%) 727 (33%) 31 (1%)

Table 8. Distribution of In(K) computations for HTW≤12.The entry in the cth row and nth column gives the numberof knots K ∈ HTWc for which it was necessary to computethe In(K) in order to guarantee classification in HTW≤12.

n\i 2 3 4 5 6 7

3 1 (100%) 1 (100%) 0 (0%) 0 (0%) 0 (0%) 0 (0%)4 1 (100%) 1 (100%) 1 (100%) 1 (100%) 0 (0%) 0 (0%)5 2 (100%) 2 (100%) 2 (100%) 2 (100%) 0 (0%) 0 (0%)

6 3 (100%) 3 (100%) 3 (100%) 2 (67%) 0 (0%) 0 (0%)7 7 (100%) 7 (100%) 7 (100%) 3 (43%) 0 (0%) 0 (0%)8 21 (100%) 21 (100%) 16 (76%) 12 (57%) 1 (5%) 0 (0%)

9 49 (100%) 49 (100%) 40 (82%) 24 (49%) 1 (2%) 0 (0%)10 165 (100%) 165 (100%) 142 (86%) 93 (56%) 6 (4%) 0 (0%)11 552 (100%) 552 (100%) 474 (86%) 301 (55%) 15 (3%) 0 (0%)

12 2176 (100%) 2176 (100%) 1692 (78%) 1004 (46%) 48 (2%) 0 (0%)13 9988 (100%) 9972 (100%) 7198 (72%) 4426 (44%) 293 (3%) 2 (0%)

Table 9. Distribution of In(K) computations for HTW≤13.The entry in the cth row and nth column gives the numberof knots K ∈ HTWc for which it was necessary to computethe In(K) in order to guarantee classification in HTW≤13.

algorithm 5.3 proposed in this paper computes a group presentation in asmall fraction of time needed for the low-index subgroup computation, areasonable strategy to minimize the total computation time is to test a fewmethods for each knot in the search of the possibly small number of criticalcells.

The problem is additionally complicated by the Tietze transformationsused to simplify the group presentation. It turns out that the number ofgenerators after transformations depends on some qualitative properties ofthe constructed discrete vector field. In Table 11 we show the number ofgenerators and relators before and after Tietze transformations in the caseof four different strategies for the construction of the discrete vector fieldfor knot K14,38437. Only in one case we get 3 generators and only in thiscase we are able to compute I7(K) in several minutes. Our conclusion


n\i 2 3 4 5 6 7

3 1 (100%) 1 (100%) 1 (100%) 1 (100%) 1 (100%) 0 (0%)4 1 (100%) 1 (100%) 1 (100%) 1 (100%) 0 (0%) 0 (0%)5 2 (100%) 2 (100%) 2 (100%) 2 (100%) 0 (0%) 0 (0%)

6 3 (100%) 3 (100%) 3 (100%) 2 (67%) 1 (33%) 0 (0%)7 7 (100%) 7 (100%) 7 (100%) 5 (71%) 2 (29%) 0 (0%)8 21 (100%) 21 (100%) 21 (100%) 13 (62%) 2 (10%) 0 (0%)

9 49 (100%) 49 (100%) 46 (94%) 26 (53%) 3 (6%) 1 (2%)10 165 (100%) 165 (100%) 158 (96%) 105 (64%) 9 (5%) 1 (1%)11 552 (100%) 552 (100%) 523 (95%) 329 (60%) 26 (5%) 0 (0%)

12 2176 (100%) 2176 (100%) 2001 (92%) 1253 (58%) 94 (4%) 0 (0%)13 9988 (100%) 9988 (100%) 8856 (89%) 5494 (55%) 451 (5%) 3 (0%)14 46972 (100%) 46934 (100%) 38092 (81%) 23634 (50%) 2231 (5%) 21 (0%)

Table 10. Distribution of In(K) computations forHTW≤14. The entry in the cth row and nth column givesthe number of knots K ∈ HTWc for which it was necessaryto compute the In(K) in order to guarantee classification inHTW≤14.

Figure 2. Knot K14,38437.

is that the number of generators does not determine the efficiency of thetransformations.

Nevertheless, for the calculations it is important to perform shaving andcollapsibleSubset steps even for knots given as arc presentation [19]. With-out the geometric simplifications we cannot get less than 4 generators forthe knot in Figure 2. The steps are also important for applications, whereknots are placed in a big cubical grid, e.g. 3D pictures of proteins.

References

[1] W. W. Boone. The word problem, Annals of Mathematics. Second Series 70 (1959),207–265


Method Before Tietze After TietzeReductions, cells order A g=4, r=4 g=4, r=3Reductions, cells order B g=4, r=4 g=3, r=2Coreductions, cells order A g=6, r=6 g=4, r=3Coreductions, cells order B g=14, r=14 g=4, r=3

Table 11. Number of generators (g) and relators (r) beforeand after Tietze transformations for knot K14,38437.

[2] P. Brendel, P. D lotko, G. Ellis, M. Juda, M. Mrozek. Computing fundamentalgroups from point clouds, Applicable Algebra in Engineering, Communication andComputing, 26(2015), 27–48.

[3] C.M. Campbell, G. Havas, E.F. Robertson. Addendum to an elementary intro-duction to coset table methods in computational group theory, London MathematicalSociety Lecture Note Series, Cambridge University Press, 71(2007), 361–364.

[4] The CAPD Group. CAPD::RedHom - Reduction Homology Algorithms:(http://redhom.ii.uj.edu.pl)

[5] M.M. Cohen. A Course in Simple-Homotopy Theory, Springer Verlag, 1973.[6] R. Engelking. General Topology, Heldermann Verlag, Berlin, 1989.[7] G. Ellis. HAP-Homological Algebra Programming, Version 1.10.13, 2013.

(http://www.gap-system.org/Packages/hap.html).[8] G.M. Fisher. On the Group of all Homeomorphisms of a Knot, Transactions of the

American Mathematical Society, 97(1960) 193–212.[9] R. Forman. Morse Theory for Cell Complexes, Advances in Mathematics, 134(1998)

90–145.[10] R. Fritsch, R. Piccinini. Cellular Structures in Topology, Cambridge University

Press, Cambridge, 1990.[11] C. McA. Gordon, J. Luecke, Knots are determined by their complements, Journal

of the American Mathematical Society, 2(1989), 371–415.[12] The GAP Group. GAP Groups, Algorithms, and Programming, Version 4.5.6, 2013.

(http://www.gap-system.org).[13] R. Geoghegan.Topological Methods in Group Theory, Springer Verlag, 2008.[14] S. Harker, K. Mischaikow, M. Mrozek, V. Nanda, H. Wagner, M. Juda, P.

D lotko. The Efficiency of a Homology Algorithm based on Discrete Morse Theoryand Coreductions, in: Proceedings of the 3rd International Workshop on Computa-tional Topology in Image Context, Chipiona, Spain, November 2010 (Rocio GonzalezDiaz and Real Jurado (Eds.)), Image A Vol. 1(2010), 41–47 (ISSN: 1885-4508)

[15] S. Harker, K. Mischaikow, M. Mrozek, V. Nanda. Discrete Morse TheoreticAlgorithms for Computing Homology of Complexes and Maps, Foundations of Com-putational Mathematics, 14(2014) 151-184, DOI:10.1007/s10208-013-9145-0.

[16] J. Hoste, M. Thistlethwaite, J. Weeks. The First 1,701,936 knots, MathematicalIntelligencer, 20(1988), 33–48.

[17] M. Joswig, M. Pfetsch. Computing optimal discrete Morse functions, ElectronicNotes in Discrete Mathematics 17 (2004), 191–195.

[18] J. Kim, M. Jin, Q.Y. Zhou, F. Luo, X. Gu. Computing fundamental group ofgeneral 3-manifold. In: Advances in Visual Computing,G. Bebis, R. Boyle, B.Parvin, D. Koracin, P. Remagnino, F. Porikli, J. Peters, J. Klosowski, L.Arns, Y. Chun, T.-M. Rhyne, L. Monroe, editors, Lecture Notes in ComputerScience, 5358(2008), 965–974.

[19] Knot Atlas, http://katlas.math.toronto.edu/wiki/Main Page


[20] D. Kozlov. Combinatorial Algebraic Topology, Springer Verlag, 2008.[21] D. Letscher. On persistent homotopy, knotted complexes and the Alexander mod-

ule, In: Proceedings of the 3rd Innovations in Theoretical Computer Science Confer-ence, ITCS ’12, ACM, New York, NY, USA, 428–441.

[22] S. Moran. The mathematical theory of knots and braids: an introduction, ElsevierScience Publishers, Amsterdam, 1983.

[23] M. Mrozek, B. Batko. Coreduction homology algorithm, Discrete and Computa-tional Geometry 41(2009), 96–118.

[24] P.S. Novikov. Ob algoritmiceskoı nerazresimosti problemy tozdestva slov v teoriigrupp, Trudy Mat. Inst. im. Steklov. 44 (1955), 143.

[25] J.H. Palmieri et al. Finite Simplicial Complexes, Sage v5.10, 2009.(http://www.sagemath.org/doc/reference/homology/sage/homology/simplicial com-plex.html).

[26] S. Rees, L.H. Soicher. An algorithmic approach to fundamental groups and coversof combinatorial cell complexes, J. Symbolic Comput., 29(2000), 59–77.

[27] E.H. Spanier. Algebraic Topology, corrected reprint of the 1966 original, Springer,New York, 1981.

[28] W.A. Stein et al. Sage Mathematics Software (Version 5.10), The Sage Develop-ment Team, 2013. (http://www.sagemath.org).

Piotr Brendel, Division of Computational Mathematics, Jagiellonian Uni-versity in Krakow, Poland. E-mail: [email protected]

Graham Ellis, School of Mathematics, National University of Ireland,Galway, Ireland. E-mail: [email protected]

Mateusz Juda, Division of Computational Mathematics, Jagiellonian Uni-versity in Krakow, Poland. E-mail: [email protected]

Marian Mrozek, Division of Computational Mathematics, Jagiellonian Uni-versity in Krakow, Poland. E-mail: [email protected]

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